According to Apollodoros,1 Zeno flourished in Ol. LXXIX. (464-460 B.C.). This date is arrived at by making him forty years younger than Parmenides, which is in direct conflict with the testimony of Plato. We have seen already (§ 84) that the meeting of Parmenides and Zeno with the young Sokrates cannot well have occurred before 449 B.C., and Plato tells us that Zeno was at that time “nearly forty years old.”2 He must, then, have been born about 489 B.C., some twenty-five years after Parmenides. He was the son of Teleutagoras, and the statement of Apollodoros that he had been adopted by Parmenides is only a misunderstanding of an expression of Plato's Sophist.3 He was, Plato further tells us,4 tall and of a graceful appearance.

Like Parmenides, Zeno played a part in the politics of his native city. Strabo, no doubt on the authority of Timaios, ascribes to him some share of the credit for the good government of Elea, and says that he was a Pythagorean.5 This statement can easily be explained. Parmenides, we have seen, was originally a Pythagorean, and the school of Elea was naturally regarded as a mere branch of the larger society. We hear also that Zeno conspired against a tyrant, whose name is differently given, and the story of his courage under torture is often repeated, though with varying details.6

Diogenes speaks of Zeno's “books,” and Souidas gives some titles which probably come from the Alexandrian librarians through Hesychios of Miletos.7 In the Parmenides Plato makes Zeno say that the work by which he is best known was written in his youth and published against his will.8 As he is supposed to be forty years old at the time of the dialogue, this must mean that the book was written before 460 B.C., and it is very possible that he wrote others after it.9 If he wrote a work against the “philosophers,” as Souidas says, that must mean the Pythagoreans, who, as we have seen, made use of the term in a sense of their own.10 The Disputations (Ἐρίδες) and the Treatise on Nature may, or may not, be the same as the book described in Plato's Parmenides.

It is not likely that Zeno wrote dialogues, though certain references in Aristotle have been supposed to imply this. In the Physics11 we hear of an argument of Zeno's, that any part of a heap of millet makes a sound, and Simplicius illustrates this by quoting a passage from a dialogue between Zeno and Protagoras.12 If our chronology is right, it is quite possible that they may have met; but it is most unlikely that Zeno should have made himself a personage in a dialogue of his own. That was a later fashion. In another place Aristotle refers to a passage where “the answerer and Zeno the questioner” occurred,13 a reference which is most easily to be understood in the same way. Alkidamas seems to have written a dialogue in which Gorgias figured,14 and the exposition of Zeno's arguments in dialogue form must always have been a tempting exercise.

Plato gives us a clear idea of what Zeno's youthful work was like. It contained more than one “discourse,” and these discourses were subdivided into sections, each dealing with some one presupposition of his adversaries.15 We owe the preservation of Zeno's arguments on the one and many to Simplicius.16 Those relating to motion have been preserved by Aristotle;17 but he has restated them in his own language.

Aristotle in his Sophist18 called Zeno the inventor of dialectic, and that, no doubt, is substantially true, though the beginnings at least of this method of arguing were contemporary with the foundation of the Eleatic school. Plato19 gives us a spirited account of the style and purpose of Zeno's book, which he puts into his own mouth:

In reality, this writing is a sort of reinforcement for the argument of Parmenides against those who try to turn it into ridicule on the ground that, if reality is one, the argument becomes involved in many absurdities and contradictions. This writing argues against those who uphold a Many, and gives them back as good and better than they gave; its aim is to show that their assumption of multiplicity will be involved in still more absurdities than the assumption of unity, if it is sufficiently worked out.

The method of Zeno was, in fact, to take one of his adversaries' fundamental postulates and deduce from it two contradictory conclusions.20 This is what Aristotle meant by calling him the inventor of dialectic, which is just the art of arguing, not from true premisses, but from premisses admitted by the other side. The theory of Parmenides had led to conclusions which contradicted the evidence of the senses, and Zeno's object was not to bring fresh proofs of the theory itself, but simply to show that his opponents' view led to contradictions of a precisely similar nature.

That Zeno's dialectic was mainly directed against the Pythagoreans is certainly suggested by Plato's statement, that it was addressed to the adversaries of Parmenides, who held that things were “a many.”21 Zeller holds, indeed that it was merely the popular form of the belief that things are many that Zeno set himself to confute;22 but it is surely not true that ordinary people believe things to be “a many” in the sense required. Plato tells us that the premisses of Zeno's arguments were the beliefs of the adversaries of Parmenides, and the postulate from which all his contradictions are derived is the view that space, and therefore body, is made up of a number of discrete units, which is just the Pythagorean doctrine. We know from Plato that Zeno's book was the work of his youth.23 It follows that he must have written it in Italy, and the Pythagoreans are the only people who can have criticised the views of Parmenides there and at that date.24

It will be noted how much clearer the historical position of Zeno becomes if we follow Plato in assigning him to a later date than is usual. We have first Parmenides, then the Pluralists, and then the criticism of Zeno. This, at any rate, seems to have been the view Aristotle took of the historical development.25

The polemic of Zeno is clearly directed in the first instance against a certain view of the unit. Eudemos, in his Physics,26 quoted from him the saying that “if any one could tell him what the unit was, he would be able to say what things are.” The commentary of Alexander on this, preserved by Simplicius, is quite satisfactory. “As Eudemos relates,” he says, “Zeno the disciple of Parmenides tried to show that it was impossible that things could be a many, seeing that there was no unit in things, whereas 'many' means a number of units.”27 Here we have a clear reference to the Pythagorean view that everything may be reduced to a sum of units, which is what Zeno denied.

The fragments of Zeno himself also show that this was his line of argument. I give them according to the arrangement of Diels.

(1)

If what is had no magnitude, it would not even be . . . . But, if it is, each one must have a certain magnitude and a certain thickness, and must be at a certain distance from another, and the same may be said of what is in front of it; for it, too, will have magnitude, and something will be in front of it.28 It is all the same to say this once and to say it always; for no such part of it will be the last, nor will one thing not be as compared with another.29

So, if things are a many, they must be both small and great, so small as not to have any magnitude at all, and so great as to be infinite. R. P. 134.

(2)

For if it were added to any other thing it would not make it any larger; for nothing can gain in magnitude by the addition of what has no magnitude, and thus it follows at once that what was added was nothing.30 But if, when this is taken away from another thing, that thing is no less; and again, if, when it is added to another thing, that does not increase, it is plain that, what was added was nothing, and what was taken away was nothing. R. P. 132.

(3)

If things are a many, they must be just as many as they are, and neither more nor less. Now, if they are as many as they are, they will be finite in number.

If things are a many, they will be infinite in number; for there will always be other things between them, and others again between these. And so things are infinite in number. R. P. 13331

161. The Unit

If we hold that the unit has no magnitude–and this is required by what Aristotle calls the argument from dichotomy,32–then everything must be infinitely small. Nothing made up of units without magnitude can itself have any magnitude. On the other hand, if we insist that the units of which things are built up are something and not nothing, we must hold that everything is infinitely great. The line is infinitely divisible; and, according to this view, it will be made up of an infinite number of units, each of which has some magnitude.

That this argument refers to points is proved by an instructive passage from Aristotle's Metaphysics.33 We read there–

If the unit is indivisible, it will, according to the proposition of Zeno, be nothing. That which neither makes anything larger by its addition to it, nor smaller by its subtraction from it, is not, he says, a real thing at all; for clearly what is real must be a magnitude. And, if it is a magnitude, it is corporeal; for that is corporeal which is in every dimension. The other things, i.e. the plane and the line, if added in one way will make things larger, added in another they will produce no effect; but the point and the unit cannot make things larger in any way.

From all this it seems impossible to draw any other conclusion than that the “one” against which Zeno argued was the “one” of which a number constitute a “many,” and that is just the Pythagorean unit.

Aristotle refers to an argument which seems to be directed against the Pythagorean doctrine of space,34 and Simplicius quotes it in this form:35

If there is space, it will be in something; for all that is is in something, and what is in something is in space. So space will be in space, and this goes on ad infinitum, therefore there is no space. R. P. 135.

What Zeno is really arguing against here is the attempt to distinguish space from the body that occupies it. If we insist that body must be in space, then we must go on to ask what space itself is in. This is a “reinforcement” of the Parmenidean denial of the void. Possibly the argument that everything must be “in” something, or must have something beyond it, had been used against the Parmenidean theory of a finite sphere with nothing outside it.

Zeno's arguments on the subject of motion have been preserved by Aristotle himself. The system of Parmenides made all motion impossible, and his successors had been driven to abandon the monistic hypothesis in order to avoid this very consequence. Zeno does not bring any fresh proofs of the impossibility of motion; all he does is to show that a pluralist theory, such as the Pythagorean, is just as unable to explain it as was that of Parmenides. Looked at in this way, Zeno's arguments are no mere quibbles, but mark a great advance in the conception of quantity. They are as follows

(1) You cannot cross a race-course.36 You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. This goes on ad infinitum, so that there are an infinite number of points in any given space, and you cannot touch an infinite number one by one in a finite time.37

(2) Achilles will never overtake the tortoise. He must first reach the place from which the tortoise started. By that time the tortoise will have got some way ahead. Achilles must then make up that, and again the tortoise will be ahead. He is always coming nearer, but he never makes up to it.38

The “hypothesis” of the second argument is the same as that of the first, namely, that the line is a series of points; but the reasoning is complicated by the introduction of another moving object. The difference, accordingly, is not a half every time, but diminishes in. a constant ratio. Again, the first argument shows that, on this hypothesis, no moving object can ever traverse any distance at all, however fast it may move; the second emphasises the fact that, however slowly it moves, it will traverse an infinite distance.39

(3) The arrow in flight is at rest. For, if everything is at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself, it cannot move.40

Here a further complication is introduced. The moving object itself has length, and its successive positions are not points but lines. The first two arguments are intended to destroy the hypothesis that a line consists of an infinite number of indivisibles; this argument and the next deal with the hypothesis that it consists of a finite41 number of indivisibles.

(4.) Half the time may be equal to double the time. Let us suppose three rows of bodies,42 one of which (A) is at rest while the other two (B, C) are moving with equal velocity in opposite directions (Fig. 1). By the time they are all in the same part of the course, B will have passed twice as many of the bodies in C as in A (Fig. 2).

Therefore the time which it takes to pass C is twice as long as the time it takes to pass A. But the time which B and C take to reach the position of A is the same. Therefore double the time is equal to the half.43

According to Aristotle, the paralogism here depends on the assumption that an equal magnitude moving with equal velocity must move for an equal time, whether the magnitude with which it is equal is at rest or in motion. That is certainly so, but we are not to suppose that this assumption is Zeno's own. The fourth argument is, in fact, related to the third just as the second is to the first. The Achilles adds a second moving point to the single moving point of the first argument; this argument adds a second moving line to the single moving line of the arrow in flight. The lines, however, are represented as a series of units, which is just how the Pythagoreans represented them; and it is quite true that, if lines are a sum of discrete units, and time is similarly a series of discrete moments, there is no other measure of motion possible than the number of units which each unit passes.

This argument, like the others, is intended to bring out the absurd conclusions which follow from the assumption that all quantity is discrete, and what Zeno has really done is to establish the conception of continuous quantity by a reductio ad absurdum of the other hypothesis. If we remember that Parmenides had asserted the one to be continuous (fr. 8, 25), we shall see how accurate is the account of Zeno's method which Plato puts into the mouth of Sokrates.

1. Diog. ix. 29 (R. P. 13o a). Apollodoros is not expressly referred to for Zeno's date; but, as he is quoted for his father's name (ix. 25; R. P. 130), there can be no doubt that he is also the source of the floruit.

2. Plato, Parm. 127 b (R. P. iii d). The visit of Zeno to Athens is confirmed by Plut. Per. 4. (R. P. 130 e), where we are told that Perikles “heard” him as well as Anaxagoras. It is also alluded to in Alc. 1. 119 a, where we are told that Pythodoros, son of Isolochos, and Kallias, son of Kalliades, each paid him 100 minae for instruction.

3. Plato, Soph. 241 d (R. P. 130 a).

4. Plato, Parm., loc. cit.

5. Strabo, vi. p. 252 (R. P. 111 c).

6. Diog. ix. 26, 27, and the other passages referred to in R. P. 130 c. The original of the account given in the tenth book of Diodoros is doubtless Timaios.

7. Diog. ix. 26 (R. P. 130); Souidas s.v. (R. P. 130 d).

8. Plato, Parm. 128 d 6 (R. P. 130 d).

9. The most remarkable title given by Souidas is Ἐξήγησις τῶν Ἐμπεδοκλέους. Of course Zeno did not write a commentary on Empedokles, but Diels points out (Berl. Sitzb., 1884, p. 359) that polemics against philosophers were sometimes called ἐξηγήσεις. Cf. the Ἡρακλείτου ἐξηγήσεις of Herakleides Pontikos and especially his Πρὸς τὸν Δημόκριτον ἐξηγήσεις (Diog. v. 88).

10. See above, p. 278, n. 1. It hardly seems likely that a later writer would make Zeno argue πρὸς τοὺς φιλοσόφους, and the title given to the book at Alexandria must be based on something contained in it.

11. Arist. Phys. H, 5. 250 a 20 (R. P. 131 a).

12. Simpl. Phys. p. 1108, 18 (R. P. 131). If this is what Aristotle refers to, it is hardly safe to attribute the κεγχρίτης λόγος to Zeno himself. The existence of this dialogue is another indication of Zeno's visit to Athens at an age when he could converse with Protagoras, which agrees very well with Plato's representation of the matter.

13. Arist. Soph. El. 170 b 22 (R. P. 130 b).

14. Chap. V. p. 199, n. 5.

15. Plato, Parm. 127 d. Plato speaks of the first ὑπόθεσις of the first λόγος, which shows that the book was really divided into separate sections. Proclus (in loc.) says there were forty of these λόγοι altogether.

16. Simplicius expressly says in one place (p. 140, 30; R. P. 133) that he is quoting κατὰ λέξιν. I see no reason to doubt this, as the Academy would certainly have a copy of the work. In that case, the use of the Attic dialect by Zeno is significant.

17. Arist. Phys. Z, 9. 239 b 9 sqq.

18. Cf. Diog. ix. 25 (R. P. 130).

19. Plato, Parm. 128 c (R. P. 130 d). If historians of philosophy had started from this careful statement of Plato's, instead of from Aristotle's loose references, they would not have failed to understand his arguments, as they all did before Tannery.

20. The technical terms used in Plato's Parmenides seem to be as old as Zeno himself. The ὑπόθεσις is the provisional assumption of the truth of a certain statement, and takes the form εἰ πολλά ἐστι or the like. The word does not mean the assumption of something as a foundation, but the setting before one's self of a statement as a problem to be solved (Ionic ὑποθέσθαι, Attic προθέσθαι). If the conclusions (τά συμβαίνοντα) which necessarily follow from the ὑπόθεσις are impossible, the ὑπόθεσις is “destroyed” (cf. Plato, Rep. 533 c 8, τὰς ὑποθέσεις ἀναιροῦσα). The author of the Περὶ ἀρχαίης ἰατρικῆς knows the word ὑπόθεσις in a similar sense.

21. The view that Zeno's arguments were directed against Pythagoreanism has been maintained in recent times by Tannery (Science hellène, pp. 249 sqq.), and Bäumker (Das Problem der Materie, pp. 60 sqq.).

22. Zeller. p. 589 (Eng. trans. p. 612).

23. Parm., loc. cit.

24. Empedokles has been suggested. He was about the same age as Zeno, indeed (§ 98), and he seems to criticise Parmenides (§ 106), but the arguments of Zeno have no special applicability to his theories. Anaxagoras is still less likely.

25. Arist. Phys. A, 3. 187 a 1 (R. P: 134 b). See below, § 173.

26. Simpl. Phys. p. 138, 32 (R. P. 134 a).

27. Simpl. Phys. p. 99, 13, ὡς γὰρ ἱστορεῖ, φησίν (Ἀλέξανδρος), Εὔδημος, Ζήνων ὁ Παρμενίδου γνώριμος ἐπειρᾶτο δεικνύναι ὅτι μὴ οἷόν τε τὰ ὄντα πολλὰ εἶναι τῷ μηδὲν εἶναι ἐν τοῖς οὖσιν ἕν, τὰ δὲ πολλὰ πλῆθος εἶναι ἑνάδων. This is the meaning of the statement that Zeno ἀνῄρει τὸ ἕν which is not Alexander's (as implied in R. P. 134 a), but goes back to no less an authority than Eudemos. It must be read in connexion with the words τὴν γὰρ στιγμὴν ὡς τὸ ἓν λέγει (Simpl. Phys. p. 99. 11).

28. I formerly rendered “the same may be said of what surpasses it in smallness; for it too will have magnitude, and something will surpass it in smallness.” This is Tannery's rendering, but I now agree with Diels in thinking that ἀπέχειν refers to μέγεθος and προέχειν to πάχος. Zeno is showing that the Pythagorean point must have three dimensions.

29. Reading, with Diels and the MSS., οὔτε ἕτερον πρὸς ἕτερον οὐκ ἔσται.. Gomperz's conjecture (adopted in R. P.) seems to me arbitrary.

30. Zeller marks a lacuna here. Zeno must certainly have shown that the subtraction of a point does not make a thing less; but he may have done so before the beginning of our present fragment.

31. This is what Aristotle calls “the argument from dichotomy” (Phys. A, 3. 187 a 2 ; R. P. 134 b). If a line is made up of points, we ought to be able to answer the question, “How many points are there in a given line?” On the other hand you can always divide a line or any part of it into two halves; so that, if a line is made up of points, there will always be more of them than any number you assign.

32. See last note.

33. Arist. Met. B, 4. 1001 b 7.

34. Arist. Phys. Δ, 1. 209 a 23; 3. 210 b 22 (R. P. 135 a).

35. Simpl. Phys. p. 562, 3 (R. P. 135). The version of Eudemos is given in Simpl. Phys. p. 563, 26, ἀξιοῖ γὰρ πᾶν τὸ ὂν ποῦ εἶναι· εἰ δὲ ὁ τόπος τῶν ὄντων, ποῦ ἂν εἴη; οὐκοῦν ἐν ἄλλῳ τόπῳ κἀκεῖνος δὴ ἐν ἄλλῳ καὶ οὕτως εἰς τὸ πρόσω..

36. Arist. Top. Θ, 8. 160 b 8, Ζήνωνος (λόγος_, ὅτι οὐκ ἐνδέχεται κινεῖσθαί οὐδὲ τὸ στάδιον διελθεῖν..

37. Arist. Phys. Z, 9, 239 b ii (R. P. 136). Cf. Z, 2. 233 a 11; a 21 (R. P., 136 a).

38. Arist. Phys. Z, 9. 239 b 14 (R. P. 137).

39. As Mr. Jourdain puts it (Mind, 1916, p. 42), “the first argument shows that motion can never begin; the second argument shows that the slower moves as fast as the faster,” on the hypothesis that a line is infinitely divisible into its constituent points.

40. Phys. Z, 9, 239 b 30 (R. P. 138); ib. 239 b 5 (R. P. 138 a). The latter passage is corrupt, though the meaning is plain. I have translated Zeller's version of it: εἰ γάρ, φησίν, ἠρεμεῖ πᾶν ὅταν ᾖ κατὰ τὸ ἴσον, ἔστι δ' ἀεὶ τὸ φερόμενον ἐν τῷ νῦν κατὰ τὸ ἴσον, ἀκίνητον κ.τ.λ.. Of course ἀεί means “at any time,” not “always,” and κατὰ τὸ ἴσον is, literally, “on a level with a space equal (to itself).” For other readings, see Zeller, p. 598 n. 3; and Diels, Vors. 19 A 27.

41. See Jourdain (loc. cit.).

42. The word is ὄγκοι; cf. Chap. VII. p. 291, n. 3. The name is very appropriate for the Pythagorean units, which Zeno had shown to have length, breadth, and thickness (fr. 1).

43. Arist. Phys. Z, 9. 239 b 33 (R. P. 139). I have had to express the argument in my own way, as it is not fully given by any of the authorities. The figure is practically Alexander's (Simpl. Phys. p. 1016, 14), except that he represents the ὄγκοι by letters instead of dots. The conclusion is plainly stated by Aristotle (loc. cit.), συμβαίνειν οἴεται ἴσον εἶναι χρόνον τῷ διπλασίῳ τὸν ἥμισυν, and, however we explain the reasoning, it must be so represented as to lead to the conclusion that, as Mr. Jourdain puts it (loc. cit.), “a body travels twice as fast as it does.”